Question
Find the zeros for the polynomial function and give the multiplicity for each zero. State whether the graph crosses the x-axis or touches the x-axis and turns around at each zero.
f(x)equals3(x minus 5)left parenthesis x minus 3 right parenthesis squared
Question content area bottom
Part 1
Determine the zero(s).
The zero(s) is/are
enter your response here.
(Type integers or decimals. Use a comma to separate answers as needed.)
Part 2
Determine the multiplicities of the zero(s). Select the correct choice below and, if necessary, fill in the answer box(es) within your choice.
A.
There is one zero. The multiplicity of the zero is
enter your response here.
(Simplify your answer.)
B.
There are two zeros. The multiplicity of the largest zero is
enter your response here. The multiplicity of the smallest zero is
enter your response here.
(Simplify your answers.)
C.
There are three zeros. The multiplicity of the largest zero is
enter your response here. The multiplicity of the smallest zero is
enter your response here. The multiplicity of the other zero is
enter your response here.
(Simplify your answers.)
Part 3
Determine the behavior of the function at each zero. Select the correct choice below and, if necessary, fill in the answer boxes within your choice.
A.
The graph crosses the x-axis at all zeros.
B.
The graph touches the x-axis and turns around at all zeros.
C.
The graph crosses the x-axis at xequals
enter your response here and touches the x-axis and turns around at xequals
enter your response here.
(Simplify your answers. Type integers or decimals. Use a comma to separate answers as needed.)
f(x)equals3(x minus 5)left parenthesis x minus 3 right parenthesis squared
Question content area bottom
Part 1
Determine the zero(s).
The zero(s) is/are
enter your response here.
(Type integers or decimals. Use a comma to separate answers as needed.)
Part 2
Determine the multiplicities of the zero(s). Select the correct choice below and, if necessary, fill in the answer box(es) within your choice.
A.
There is one zero. The multiplicity of the zero is
enter your response here.
(Simplify your answer.)
B.
There are two zeros. The multiplicity of the largest zero is
enter your response here. The multiplicity of the smallest zero is
enter your response here.
(Simplify your answers.)
C.
There are three zeros. The multiplicity of the largest zero is
enter your response here. The multiplicity of the smallest zero is
enter your response here. The multiplicity of the other zero is
enter your response here.
(Simplify your answers.)
Part 3
Determine the behavior of the function at each zero. Select the correct choice below and, if necessary, fill in the answer boxes within your choice.
A.
The graph crosses the x-axis at all zeros.
B.
The graph touches the x-axis and turns around at all zeros.
C.
The graph crosses the x-axis at xequals
enter your response here and touches the x-axis and turns around at xequals
enter your response here.
(Simplify your answers. Type integers or decimals. Use a comma to separate answers as needed.)
Answers
GPT-4o mini
To find the zeros, multiplicities, and behavior of the polynomial function \( f(x) = 3(x - 5)(x - 3)^2 \), let’s break it down step by step.
### Part 1: Determine the zeros.
To find the zeros, we set \( f(x) = 0 \):
\[
3(x - 5)(x - 3)^2 = 0
\]
This gives us two factors to solve:
1. \( x - 5 = 0 \)
Thus, \( x = 5 \)
2. \( (x - 3)^2 = 0 \)
Thus, \( x - 3 = 0 \), giving \( x = 3 \)
The zeros of the function are:
- \( 5 \)
- \( 3 \)
So, the zeros are:
\[
\text{The zeros are } 3, 5.
\]
### Part 2: Determine the multiplicities of the zeros.
- The zero \( x = 5 \) comes from the factor \( (x - 5) \), which has a multiplicity of \( 1 \).
- The zero \( x = 3 \) comes from the factor \( (x - 3)^2 \), which has a multiplicity of \( 2 \).
Thus, we have:
- For \( x = 5 \): multiplicity \( 1 \)
- For \( x = 3 \): multiplicity \( 2 \)
The \( B \) option is correct here:
\[
\text{There are two zeros. The multiplicity of the largest zero (5) is } 1 \text{. The multiplicity of the smallest zero (3) is } 2.
\]
### Part 3: Determine the behavior of the function at each zero.
1. **For zero \( x = 5 \)** (multiplicity 1): Since it has an odd multiplicity, the graph will cross the x-axis at this point.
2. **For zero \( x = 3 \)** (multiplicity 2): Since it has an even multiplicity, the graph will touch the x-axis and turn around at this point.
Combining this information:
\[
\text{The graph crosses the x-axis at } x = 5 \text{ and touches the x-axis and turns around at } x = 3.
\]
Thus, the behavior of the function is:
\[
\text{C. The graph crosses the x-axis at } x = 5 \text{ and touches the x-axis and turns around at } x = 3.
\]
### Final Results
**Part 1:**
- The zeros are
\( 3, 5 \)
**Part 2:**
- There are two zeros. The multiplicity of the largest zero is
\( 1 \). The multiplicity of the smallest zero is
\( 2 \).
**Part 3:**
- The graph crosses the x-axis at \( 5 \) and touches the x-axis and turns around at \( 3 \).
### Part 1: Determine the zeros.
To find the zeros, we set \( f(x) = 0 \):
\[
3(x - 5)(x - 3)^2 = 0
\]
This gives us two factors to solve:
1. \( x - 5 = 0 \)
Thus, \( x = 5 \)
2. \( (x - 3)^2 = 0 \)
Thus, \( x - 3 = 0 \), giving \( x = 3 \)
The zeros of the function are:
- \( 5 \)
- \( 3 \)
So, the zeros are:
\[
\text{The zeros are } 3, 5.
\]
### Part 2: Determine the multiplicities of the zeros.
- The zero \( x = 5 \) comes from the factor \( (x - 5) \), which has a multiplicity of \( 1 \).
- The zero \( x = 3 \) comes from the factor \( (x - 3)^2 \), which has a multiplicity of \( 2 \).
Thus, we have:
- For \( x = 5 \): multiplicity \( 1 \)
- For \( x = 3 \): multiplicity \( 2 \)
The \( B \) option is correct here:
\[
\text{There are two zeros. The multiplicity of the largest zero (5) is } 1 \text{. The multiplicity of the smallest zero (3) is } 2.
\]
### Part 3: Determine the behavior of the function at each zero.
1. **For zero \( x = 5 \)** (multiplicity 1): Since it has an odd multiplicity, the graph will cross the x-axis at this point.
2. **For zero \( x = 3 \)** (multiplicity 2): Since it has an even multiplicity, the graph will touch the x-axis and turn around at this point.
Combining this information:
\[
\text{The graph crosses the x-axis at } x = 5 \text{ and touches the x-axis and turns around at } x = 3.
\]
Thus, the behavior of the function is:
\[
\text{C. The graph crosses the x-axis at } x = 5 \text{ and touches the x-axis and turns around at } x = 3.
\]
### Final Results
**Part 1:**
- The zeros are
\( 3, 5 \)
**Part 2:**
- There are two zeros. The multiplicity of the largest zero is
\( 1 \). The multiplicity of the smallest zero is
\( 2 \).
**Part 3:**
- The graph crosses the x-axis at \( 5 \) and touches the x-axis and turns around at \( 3 \).